1. Split input into 3 regimes

2. if b < -4.39999999999999995e-71

   1. Initial program 13.4%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

   2. Taylor expanded in b around -inf 82.0%

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   3. Step-by-step derivation

      1. associate-*r/82.0%

         \[\leadsto \color{blue}{\frac{-1 \cdot c}{b}} \]

      2. neg-mul-182.0%

         \[\leadsto \frac{\color{blue}{-c}}{b} \]

   4. Simplified82.0%

      \[\leadsto \color{blue}{\frac{-c}{b}} \]

   if -4.39999999999999995e-71 < b < 3.8e14

   1. Initial program 79.6%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

   if 3.8e14 < b

   1. Initial program 66.1%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

   2. Taylor expanded in b around inf 98.8%

      \[\leadsto \color{blue}{\frac{c}{b} + -1 \cdot \frac{b}{a}} \]
   3. Step-by-step derivation

      1. mul-1-neg98.8%

         \[\leadsto \frac{c}{b} + \color{blue}{\left(-\frac{b}{a}\right)} \]

      2. unsub-neg98.8%

         \[\leadsto \color{blue}{\frac{c}{b} - \frac{b}{a}} \]

   4. Simplified98.8%

      \[\leadsto \color{blue}{\frac{c}{b} - \frac{b}{a}} \]
3. Recombined 3 regimes into one program.

4. Final simplification85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.4 \cdot 10^{-71}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 3.8 \cdot 10^{+14}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

1. Split input into 3 regimes

2. if b < -1.10000000000000001e-72

   1. Initial program 13.4%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

   2. Taylor expanded in b around -inf 82.0%

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   3. Step-by-step derivation

      1. associate-*r/82.0%

         \[\leadsto \color{blue}{\frac{-1 \cdot c}{b}} \]

      2. neg-mul-182.0%

         \[\leadsto \frac{\color{blue}{-c}}{b} \]

   4. Simplified82.0%

      \[\leadsto \color{blue}{\frac{-c}{b}} \]

   if -1.10000000000000001e-72 < b < 1.65000000000000004e-21

   1. Initial program 76.9%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. /-rgt-identity76.9%

         \[\leadsto \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{\frac{2 \cdot a}{1}}} \]

      2. metadata-eval76.9%

         \[\leadsto \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\frac{2 \cdot a}{\color{blue}{--1}}} \]

      3. associate-/l*76.9%

         \[\leadsto \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{\frac{2}{\frac{--1}{a}}}} \]

      4. associate-/r/76.9%

         \[\leadsto \color{blue}{\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2} \cdot \frac{--1}{a}} \]

      5. *-commutative76.9%

         \[\leadsto \color{blue}{\frac{--1}{a} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2}} \]

      6. metadata-eval76.9%

         \[\leadsto \frac{\color{blue}{1}}{a} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2} \]

      7. metadata-eval76.9%

         \[\leadsto \frac{\color{blue}{-1 \cdot -1}}{a} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2} \]

      8. associate-*l/76.9%

         \[\leadsto \color{blue}{\left(\frac{-1}{a} \cdot -1\right)} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2} \]

      9. associate-/r/76.9%

         \[\leadsto \color{blue}{\frac{-1}{\frac{a}{-1}}} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2} \]

      10. times-frac76.9%

          \[\leadsto \color{blue}{\frac{-1 \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}{\frac{a}{-1} \cdot 2}} \]

      11. *-commutative76.9%

          \[\leadsto \frac{-1 \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}{\color{blue}{2 \cdot \frac{a}{-1}}} \]

      12. times-frac76.9%

          \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\frac{a}{-1}}} \]

      13. metadata-eval76.9%

          \[\leadsto \color{blue}{-0.5} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\frac{a}{-1}} \]

      14. associate-/r/76.9%

          \[\leadsto -0.5 \cdot \color{blue}{\left(\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a} \cdot -1\right)} \]

      15. *-commutative76.9%

          \[\leadsto -0.5 \cdot \color{blue}{\left(-1 \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a}\right)} \]

      16. div-sub77.0%

          \[\leadsto -0.5 \cdot \left(-1 \cdot \color{blue}{\left(\frac{-b}{a} - \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a}\right)}\right) \]

   3. Simplified76.9%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}{a}} \]
   4. Step-by-step derivation

      1. add-sqr-sqrt76.6%

         \[\leadsto -0.5 \cdot \frac{b + \color{blue}{\sqrt{\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}} \cdot \sqrt{\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}}}{a} \]

      2. pow276.6%

         \[\leadsto -0.5 \cdot \frac{b + \color{blue}{{\left(\sqrt{\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}\right)}^{2}}}{a} \]

      3. pow1/276.6%

         \[\leadsto -0.5 \cdot \frac{b + {\left(\sqrt{\color{blue}{{\left(\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)\right)}^{0.5}}}\right)}^{2}}{a} \]

      4. sqrt-pow176.6%

         \[\leadsto -0.5 \cdot \frac{b + {\color{blue}{\left({\left(\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}}^{2}}{a} \]

      5. metadata-eval76.6%

         \[\leadsto -0.5 \cdot \frac{b + {\left({\left(\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)\right)}^{\color{blue}{0.25}}\right)}^{2}}{a} \]

   5. Applied egg-rr76.6%

      \[\leadsto -0.5 \cdot \frac{b + \color{blue}{{\left({\left(\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)\right)}^{0.25}\right)}^{2}}}{a} \]

   6. Taylor expanded in b around 0 69.7%

      \[\leadsto -0.5 \cdot \frac{b + {\color{blue}{\left({\left(-4 \cdot \left(c \cdot a\right)\right)}^{0.25}\right)}}^{2}}{a} \]
   7. Step-by-step derivation

      1. *-commutative69.7%

         \[\leadsto -0.5 \cdot \frac{b + {\left({\color{blue}{\left(\left(c \cdot a\right) \cdot -4\right)}}^{0.25}\right)}^{2}}{a} \]

   8. Simplified69.7%

      \[\leadsto -0.5 \cdot \frac{b + {\color{blue}{\left({\left(\left(c \cdot a\right) \cdot -4\right)}^{0.25}\right)}}^{2}}{a} \]

   9. Taylor expanded in c around 0 39.3%

      \[\leadsto -0.5 \cdot \frac{b + \color{blue}{{\left(e^{0.25 \cdot \left(\log \left(-4 \cdot a\right) + \log c\right)}\right)}^{2}}}{a} \]
   10. Step-by-step derivation

       1. +-commutative39.3%

          \[\leadsto -0.5 \cdot \frac{b + {\left(e^{0.25 \cdot \color{blue}{\left(\log c + \log \left(-4 \cdot a\right)\right)}}\right)}^{2}}{a} \]

       2. *-commutative39.3%

          \[\leadsto -0.5 \cdot \frac{b + {\left(e^{0.25 \cdot \left(\log c + \log \color{blue}{\left(a \cdot -4\right)}\right)}\right)}^{2}}{a} \]

       3. log-prod65.9%

          \[\leadsto -0.5 \cdot \frac{b + {\left(e^{0.25 \cdot \color{blue}{\log \left(c \cdot \left(a \cdot -4\right)\right)}}\right)}^{2}}{a} \]

       4. *-commutative65.9%

          \[\leadsto -0.5 \cdot \frac{b + {\left(e^{\color{blue}{\log \left(c \cdot \left(a \cdot -4\right)\right) \cdot 0.25}}\right)}^{2}}{a} \]

       5. exp-to-pow69.7%

          \[\leadsto -0.5 \cdot \frac{b + {\color{blue}{\left({\left(c \cdot \left(a \cdot -4\right)\right)}^{0.25}\right)}}^{2}}{a} \]

       6. unpow269.7%

          \[\leadsto -0.5 \cdot \frac{b + \color{blue}{{\left(c \cdot \left(a \cdot -4\right)\right)}^{0.25} \cdot {\left(c \cdot \left(a \cdot -4\right)\right)}^{0.25}}}{a} \]

       7. pow-sqr70.0%

          \[\leadsto -0.5 \cdot \frac{b + \color{blue}{{\left(c \cdot \left(a \cdot -4\right)\right)}^{\left(2 \cdot 0.25\right)}}}{a} \]

       8. metadata-eval70.0%

          \[\leadsto -0.5 \cdot \frac{b + {\left(c \cdot \left(a \cdot -4\right)\right)}^{\color{blue}{0.5}}}{a} \]

       9. unpow1/270.0%

          \[\leadsto -0.5 \cdot \frac{b + \color{blue}{\sqrt{c \cdot \left(a \cdot -4\right)}}}{a} \]

   11. Simplified70.0%

       \[\leadsto -0.5 \cdot \frac{b + \color{blue}{\sqrt{c \cdot \left(a \cdot -4\right)}}}{a} \]

   if 1.65000000000000004e-21 < b

   1. Initial program 70.9%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

   2. Taylor expanded in b around inf 91.5%

      \[\leadsto \frac{\color{blue}{2 \cdot \frac{c \cdot a}{b} + -2 \cdot b}}{2 \cdot a} \]
   3. Step-by-step derivation

      1. +-commutative91.5%

         \[\leadsto \frac{\color{blue}{-2 \cdot b + 2 \cdot \frac{c \cdot a}{b}}}{2 \cdot a} \]

      2. *-commutative91.5%

         \[\leadsto \frac{\color{blue}{b \cdot -2} + 2 \cdot \frac{c \cdot a}{b}}{2 \cdot a} \]

      3. fma-def91.5%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(b, -2, 2 \cdot \frac{c \cdot a}{b}\right)}}{2 \cdot a} \]

      4. *-commutative91.5%

         \[\leadsto \frac{\mathsf{fma}\left(b, -2, \color{blue}{\frac{c \cdot a}{b} \cdot 2}\right)}{2 \cdot a} \]

      5. associate-/l*95.3%

         \[\leadsto \frac{\mathsf{fma}\left(b, -2, \color{blue}{\frac{c}{\frac{b}{a}}} \cdot 2\right)}{2 \cdot a} \]

   4. Simplified95.3%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(b, -2, \frac{c}{\frac{b}{a}} \cdot 2\right)}}{2 \cdot a} \]
3. Recombined 3 regimes into one program.

4. Final simplification82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.1 \cdot 10^{-72}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 1.65 \cdot 10^{-21}:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{c \cdot \left(a \cdot -4\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, -2, 2 \cdot \frac{c}{\frac{b}{a}}\right)}{a \cdot 2}\\ \end{array} \]

1. Split input into 3 regimes

2. if b < -5.79999999999999995e-72

   1. Initial program 13.4%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

   2. Taylor expanded in b around -inf 82.0%

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   3. Step-by-step derivation

      1. associate-*r/82.0%

         \[\leadsto \color{blue}{\frac{-1 \cdot c}{b}} \]

      2. neg-mul-182.0%

         \[\leadsto \frac{\color{blue}{-c}}{b} \]

   4. Simplified82.0%

      \[\leadsto \color{blue}{\frac{-c}{b}} \]

   if -5.79999999999999995e-72 < b < 2.05e-20

   1. Initial program 76.9%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. /-rgt-identity76.9%

         \[\leadsto \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{\frac{2 \cdot a}{1}}} \]

      2. metadata-eval76.9%

         \[\leadsto \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\frac{2 \cdot a}{\color{blue}{--1}}} \]

      3. associate-/l*76.9%

         \[\leadsto \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{\frac{2}{\frac{--1}{a}}}} \]

      4. associate-/r/76.9%

         \[\leadsto \color{blue}{\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2} \cdot \frac{--1}{a}} \]

      5. *-commutative76.9%

         \[\leadsto \color{blue}{\frac{--1}{a} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2}} \]

      6. metadata-eval76.9%

         \[\leadsto \frac{\color{blue}{1}}{a} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2} \]

      7. metadata-eval76.9%

         \[\leadsto \frac{\color{blue}{-1 \cdot -1}}{a} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2} \]

      8. associate-*l/76.9%

         \[\leadsto \color{blue}{\left(\frac{-1}{a} \cdot -1\right)} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2} \]

      9. associate-/r/76.9%

         \[\leadsto \color{blue}{\frac{-1}{\frac{a}{-1}}} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2} \]

      10. times-frac76.9%

          \[\leadsto \color{blue}{\frac{-1 \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}{\frac{a}{-1} \cdot 2}} \]

      11. *-commutative76.9%

          \[\leadsto \frac{-1 \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}{\color{blue}{2 \cdot \frac{a}{-1}}} \]

      12. times-frac76.9%

          \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\frac{a}{-1}}} \]

      13. metadata-eval76.9%

          \[\leadsto \color{blue}{-0.5} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\frac{a}{-1}} \]

      14. associate-/r/76.9%

          \[\leadsto -0.5 \cdot \color{blue}{\left(\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a} \cdot -1\right)} \]

      15. *-commutative76.9%

          \[\leadsto -0.5 \cdot \color{blue}{\left(-1 \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a}\right)} \]

      16. div-sub77.0%

          \[\leadsto -0.5 \cdot \left(-1 \cdot \color{blue}{\left(\frac{-b}{a} - \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a}\right)}\right) \]

   3. Simplified76.9%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}{a}} \]
   4. Step-by-step derivation

      1. add-sqr-sqrt76.6%

         \[\leadsto -0.5 \cdot \frac{b + \color{blue}{\sqrt{\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}} \cdot \sqrt{\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}}}{a} \]

      2. pow276.6%

         \[\leadsto -0.5 \cdot \frac{b + \color{blue}{{\left(\sqrt{\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}\right)}^{2}}}{a} \]

      3. pow1/276.6%

         \[\leadsto -0.5 \cdot \frac{b + {\left(\sqrt{\color{blue}{{\left(\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)\right)}^{0.5}}}\right)}^{2}}{a} \]

      4. sqrt-pow176.6%

         \[\leadsto -0.5 \cdot \frac{b + {\color{blue}{\left({\left(\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}}^{2}}{a} \]

      5. metadata-eval76.6%

         \[\leadsto -0.5 \cdot \frac{b + {\left({\left(\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)\right)}^{\color{blue}{0.25}}\right)}^{2}}{a} \]

   5. Applied egg-rr76.6%

      \[\leadsto -0.5 \cdot \frac{b + \color{blue}{{\left({\left(\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)\right)}^{0.25}\right)}^{2}}}{a} \]

   6. Taylor expanded in b around 0 69.7%

      \[\leadsto -0.5 \cdot \frac{b + {\color{blue}{\left({\left(-4 \cdot \left(c \cdot a\right)\right)}^{0.25}\right)}}^{2}}{a} \]
   7. Step-by-step derivation

      1. *-commutative69.7%

         \[\leadsto -0.5 \cdot \frac{b + {\left({\color{blue}{\left(\left(c \cdot a\right) \cdot -4\right)}}^{0.25}\right)}^{2}}{a} \]

   8. Simplified69.7%

      \[\leadsto -0.5 \cdot \frac{b + {\color{blue}{\left({\left(\left(c \cdot a\right) \cdot -4\right)}^{0.25}\right)}}^{2}}{a} \]

   9. Taylor expanded in c around 0 39.3%

      \[\leadsto -0.5 \cdot \frac{b + \color{blue}{{\left(e^{0.25 \cdot \left(\log \left(-4 \cdot a\right) + \log c\right)}\right)}^{2}}}{a} \]
   10. Step-by-step derivation

       1. +-commutative39.3%

          \[\leadsto -0.5 \cdot \frac{b + {\left(e^{0.25 \cdot \color{blue}{\left(\log c + \log \left(-4 \cdot a\right)\right)}}\right)}^{2}}{a} \]

       2. *-commutative39.3%

          \[\leadsto -0.5 \cdot \frac{b + {\left(e^{0.25 \cdot \left(\log c + \log \color{blue}{\left(a \cdot -4\right)}\right)}\right)}^{2}}{a} \]

       3. log-prod65.9%

          \[\leadsto -0.5 \cdot \frac{b + {\left(e^{0.25 \cdot \color{blue}{\log \left(c \cdot \left(a \cdot -4\right)\right)}}\right)}^{2}}{a} \]

       4. *-commutative65.9%

          \[\leadsto -0.5 \cdot \frac{b + {\left(e^{\color{blue}{\log \left(c \cdot \left(a \cdot -4\right)\right) \cdot 0.25}}\right)}^{2}}{a} \]

       5. exp-to-pow69.7%

          \[\leadsto -0.5 \cdot \frac{b + {\color{blue}{\left({\left(c \cdot \left(a \cdot -4\right)\right)}^{0.25}\right)}}^{2}}{a} \]

       6. unpow269.7%

          \[\leadsto -0.5 \cdot \frac{b + \color{blue}{{\left(c \cdot \left(a \cdot -4\right)\right)}^{0.25} \cdot {\left(c \cdot \left(a \cdot -4\right)\right)}^{0.25}}}{a} \]

       7. pow-sqr70.0%

          \[\leadsto -0.5 \cdot \frac{b + \color{blue}{{\left(c \cdot \left(a \cdot -4\right)\right)}^{\left(2 \cdot 0.25\right)}}}{a} \]

       8. metadata-eval70.0%

          \[\leadsto -0.5 \cdot \frac{b + {\left(c \cdot \left(a \cdot -4\right)\right)}^{\color{blue}{0.5}}}{a} \]

       9. unpow1/270.0%

          \[\leadsto -0.5 \cdot \frac{b + \color{blue}{\sqrt{c \cdot \left(a \cdot -4\right)}}}{a} \]

   11. Simplified70.0%

       \[\leadsto -0.5 \cdot \frac{b + \color{blue}{\sqrt{c \cdot \left(a \cdot -4\right)}}}{a} \]

   if 2.05e-20 < b

   1. Initial program 70.9%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

   2. Taylor expanded in b around inf 95.3%

      \[\leadsto \color{blue}{\frac{c}{b} + -1 \cdot \frac{b}{a}} \]
   3. Step-by-step derivation

      1. mul-1-neg95.3%

         \[\leadsto \frac{c}{b} + \color{blue}{\left(-\frac{b}{a}\right)} \]

      2. unsub-neg95.3%

         \[\leadsto \color{blue}{\frac{c}{b} - \frac{b}{a}} \]

   4. Simplified95.3%

      \[\leadsto \color{blue}{\frac{c}{b} - \frac{b}{a}} \]
3. Recombined 3 regimes into one program.

4. Final simplification82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5.8 \cdot 10^{-72}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 2.05 \cdot 10^{-20}:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{c \cdot \left(a \cdot -4\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 29.9%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

   2. Taylor expanded in b around -inf 63.8%

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   3. Step-by-step derivation

      1. associate-*r/63.8%

         \[\leadsto \color{blue}{\frac{-1 \cdot c}{b}} \]

      2. neg-mul-163.8%

         \[\leadsto \frac{\color{blue}{-c}}{b} \]

   4. Simplified63.8%

      \[\leadsto \color{blue}{\frac{-c}{b}} \]

   if -4.999999999999985e-310 < b

   1. Initial program 75.9%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

   2. Taylor expanded in b around inf 67.7%

      \[\leadsto \color{blue}{\frac{c}{b} + -1 \cdot \frac{b}{a}} \]
   3. Step-by-step derivation

      1. mul-1-neg67.7%

         \[\leadsto \frac{c}{b} + \color{blue}{\left(-\frac{b}{a}\right)} \]

      2. unsub-neg67.7%

         \[\leadsto \color{blue}{\frac{c}{b} - \frac{b}{a}} \]

   4. Simplified67.7%

      \[\leadsto \color{blue}{\frac{c}{b} - \frac{b}{a}} \]
3. Recombined 2 regimes into one program.

4. Final simplification65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

1. Split input into 2 regimes

2. if b < -1.50000000000000003e88

   1. Initial program 7.5%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

   2. Taylor expanded in b around inf 2.2%

      \[\leadsto \frac{\color{blue}{2 \cdot \frac{c \cdot a}{b} + -2 \cdot b}}{2 \cdot a} \]
   3. Step-by-step derivation

      1. +-commutative2.2%

         \[\leadsto \frac{\color{blue}{-2 \cdot b + 2 \cdot \frac{c \cdot a}{b}}}{2 \cdot a} \]

      2. *-commutative2.2%

         \[\leadsto \frac{\color{blue}{b \cdot -2} + 2 \cdot \frac{c \cdot a}{b}}{2 \cdot a} \]

      3. fma-def2.2%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(b, -2, 2 \cdot \frac{c \cdot a}{b}\right)}}{2 \cdot a} \]

      4. *-commutative2.2%

         \[\leadsto \frac{\mathsf{fma}\left(b, -2, \color{blue}{\frac{c \cdot a}{b} \cdot 2}\right)}{2 \cdot a} \]

      5. associate-/l*2.5%

         \[\leadsto \frac{\mathsf{fma}\left(b, -2, \color{blue}{\frac{c}{\frac{b}{a}}} \cdot 2\right)}{2 \cdot a} \]

   4. Simplified2.5%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(b, -2, \frac{c}{\frac{b}{a}} \cdot 2\right)}}{2 \cdot a} \]

   5. Taylor expanded in b around 0 27.1%

      \[\leadsto \color{blue}{\frac{c}{b}} \]

   if -1.50000000000000003e88 < b

   1. Initial program 65.3%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

   2. Taylor expanded in b around inf 43.1%

      \[\leadsto \color{blue}{-1 \cdot \frac{b}{a}} \]
   3. Step-by-step derivation

      1. associate-*r/43.1%

         \[\leadsto \color{blue}{\frac{-1 \cdot b}{a}} \]

      2. mul-1-neg43.1%

         \[\leadsto \frac{\color{blue}{-b}}{a} \]

   4. Simplified43.1%

      \[\leadsto \color{blue}{\frac{-b}{a}} \]
3. Recombined 2 regimes into one program.

4. Final simplification39.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.5 \cdot 10^{+88}:\\ \;\;\;\;\frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;-\frac{b}{a}\\ \end{array} \]

1. Split input into 2 regimes

2. if b < -3.4999999999999998e-305

   1. Initial program 29.3%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

   2. Taylor expanded in b around -inf 64.3%

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   3. Step-by-step derivation

      1. associate-*r/64.3%

         \[\leadsto \color{blue}{\frac{-1 \cdot c}{b}} \]

      2. neg-mul-164.3%

         \[\leadsto \frac{\color{blue}{-c}}{b} \]

   4. Simplified64.3%

      \[\leadsto \color{blue}{\frac{-c}{b}} \]

   if -3.4999999999999998e-305 < b

   1. Initial program 76.1%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

   2. Taylor expanded in b around inf 66.8%

      \[\leadsto \color{blue}{-1 \cdot \frac{b}{a}} \]
   3. Step-by-step derivation

      1. associate-*r/66.8%

         \[\leadsto \color{blue}{\frac{-1 \cdot b}{a}} \]

      2. mul-1-neg66.8%

         \[\leadsto \frac{\color{blue}{-b}}{a} \]

   4. Simplified66.8%

      \[\leadsto \color{blue}{\frac{-b}{a}} \]
3. Recombined 2 regimes into one program.

4. Final simplification65.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.5 \cdot 10^{-305}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;-\frac{b}{a}\\ \end{array} \]

1. Initial program 52.0%

   \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
2. Step-by-step derivation

   1. clear-num51.9%

      \[\leadsto \color{blue}{\frac{1}{\frac{2 \cdot a}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}} \]

   2. associate-/r/51.9%

      \[\leadsto \color{blue}{\frac{1}{2 \cdot a} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)} \]

   3. associate-/r*51.9%

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{a}} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \]

   4. metadata-eval51.9%

      \[\leadsto \frac{\color{blue}{0.5}}{a} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \]

   5. add-sqr-sqrt50.9%

      \[\leadsto \frac{0.5}{a} \cdot \left(\left(-b\right) - \color{blue}{\sqrt{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}} \cdot \sqrt{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\right) \]

   6. cancel-sign-sub-inv50.9%

      \[\leadsto \frac{0.5}{a} \cdot \color{blue}{\left(\left(-b\right) + \left(-\sqrt{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\right) \cdot \sqrt{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\right)} \]

   7. add-sqr-sqrt15.5%

      \[\leadsto \frac{0.5}{a} \cdot \left(\color{blue}{\sqrt{-b} \cdot \sqrt{-b}} + \left(-\sqrt{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\right) \cdot \sqrt{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\right) \]

   8. sqrt-unprod29.0%

      \[\leadsto \frac{0.5}{a} \cdot \left(\color{blue}{\sqrt{\left(-b\right) \cdot \left(-b\right)}} + \left(-\sqrt{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\right) \cdot \sqrt{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\right) \]

   9. sqr-neg29.0%

      \[\leadsto \frac{0.5}{a} \cdot \left(\sqrt{\color{blue}{b \cdot b}} + \left(-\sqrt{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\right) \cdot \sqrt{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\right) \]

   10. sqrt-prod21.1%

       \[\leadsto \frac{0.5}{a} \cdot \left(\color{blue}{\sqrt{b} \cdot \sqrt{b}} + \left(-\sqrt{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\right) \cdot \sqrt{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\right) \]

   11. add-sqr-sqrt35.5%

       \[\leadsto \frac{0.5}{a} \cdot \left(\color{blue}{b} + \left(-\sqrt{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\right) \cdot \sqrt{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\right) \]

3. Applied egg-rr34.7%

   \[\leadsto \color{blue}{\frac{0.5}{a} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right)} \]

4. Taylor expanded in b around -inf 2.4%

   \[\leadsto \color{blue}{\frac{b}{a}} \]

5. Final simplification2.4%

   \[\leadsto \frac{b}{a} \]

1. Initial program 52.0%

   \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

2. Taylor expanded in b around inf 32.4%

   \[\leadsto \frac{\color{blue}{2 \cdot \frac{c \cdot a}{b} + -2 \cdot b}}{2 \cdot a} \]
3. Step-by-step derivation

   1. +-commutative32.4%

      \[\leadsto \frac{\color{blue}{-2 \cdot b + 2 \cdot \frac{c \cdot a}{b}}}{2 \cdot a} \]

   2. *-commutative32.4%

      \[\leadsto \frac{\color{blue}{b \cdot -2} + 2 \cdot \frac{c \cdot a}{b}}{2 \cdot a} \]

   3. fma-def32.4%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(b, -2, 2 \cdot \frac{c \cdot a}{b}\right)}}{2 \cdot a} \]

   4. *-commutative32.4%

      \[\leadsto \frac{\mathsf{fma}\left(b, -2, \color{blue}{\frac{c \cdot a}{b} \cdot 2}\right)}{2 \cdot a} \]

   5. associate-/l*33.7%

      \[\leadsto \frac{\mathsf{fma}\left(b, -2, \color{blue}{\frac{c}{\frac{b}{a}}} \cdot 2\right)}{2 \cdot a} \]

4. Simplified33.7%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(b, -2, \frac{c}{\frac{b}{a}} \cdot 2\right)}}{2 \cdot a} \]

5. Taylor expanded in b around 0 8.4%

   \[\leadsto \color{blue}{\frac{c}{b}} \]

6. Final simplification8.4%

   \[\leadsto \frac{c}{b} \]

herbie shell --seed 2023166 
(FPCore (a b c)
  :name "quadm (p42, negative)"
  :precision binary64

  :herbie-target
  (if (< b 0.0) (/ c (* a (/ (+ (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))) (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))

  (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))